Functions in Context

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Answer Key & Explanations

Functions in Context · Grade 9 · Worksheet 1

1. Liam is designing a parabolic arch for a bridge. The arch's height above the ground is modeled by the function h(x) = -0.02x² + 1.2x, where x is the horizontal distance in meters from the left support and h(x) is the height in meters. What is the maximum height of the arch above the ground? Answer: 18 Solution: h(x) = -0.02x² + 1.2x This is a quadratic function in the form ax² + bx + c, where: a = -0.02 b = 1.2 c = 0 Since a < 0, the parabola opens downward, so the vertex gives the maximum height.
   Full step-by-step solution

   We are given the function for the arch height: h(x) = -0.02x² + 1.2x This is a quadratic function in the form ax² + bx + c, where: a = -0.02 b = 1.2 c = 0 Since a < 0, the parabola opens downward, so the vertex gives the maximum height. The x-coordinate of the vertex is: x = -b / (2a) Substitute b and a: x = -1.2 / (2 * -0.02) x = -1.2 / (-0.04) x = 1.2 / 0.04 x = 120 / 4 x = 30 So the maximum height occurs at x = 30 meters. Now substitute x = 30 into h(x): h(30) = -0.02 * (30)² + 1.2 * 30 First compute (30)² = 900. Then -0.02 * 900 = -18. Then 1.2 * 30 = 36. So h(30) = -18 + 36 = 18. Therefore, the maximum height of the arch is 18 meters.

2. Noah is tracking the height of a rocket launched from a platform. The height h(t) in meters after t seconds is modeled by the function h(t) = -5t² + 41t + 6. What does the value h(6) represent in this context? Answer: The height of the rocket 6 seconds after launch is 72 meters. Solution: The function h(t) = -5t² + 41t + 6 models the rocket's height in meters at time t seconds.
   Full step-by-step solution

   Step 1: The function h(t) = -5t² + 41t + 6 models the rocket's height in meters at time t seconds. Step 2: To find h(6), substitute t = 6 into the function: h(6) = -5(6)² + 41(6) + 6 Step 3: Calculate the square: (6)² = 36, so -5(36) = -180 Step 4: Multiply: 41(6) = 246 Step 5: Add the terms: h(6) = -180 + 246 + 6 = 72 Step 6: Interpret the result: h(6) = 72 means that 6 seconds after launch, the rocket is 72 meters above the ground. The answer is: The height of the rocket 6 seconds after launch is 72 meters.

3. Matiu is analyzing the water level in a reservoir during a drought. The water volume V(t) in millions of liters is modeled by the quadratic function V(t) = -0.4t² + 8t + 120, where t represents the number of days since the drought began. Interpret the meaning of V(15) and calculate its value. Answer: V(15) = 150; after 15 days, the reservoir contains 150 million liters of water. Solution: The function V(t) = -0.4t² + 8t + 120 models the water volume in millions of liters after t days. V(15) means we evaluate the function at t = 15, which gives the water volume on day 15.
   Full step-by-step solution

   Step 1: The function V(t) = -0.4t² + 8t + 120 models the water volume in millions of liters after t days. V(15) means we evaluate the function at t = 15, which gives the water volume on day 15. Step 2: Substitute t = 15 into the function: V(15) = -0.4(15)² + 8(15) + 120. Step 3: Calculate 15² = 225. Step 4: Multiply: -0.4 × 225 = -90. Step 5: Multiply: 8 × 15 = 120. Step 6: Add the terms: -90 + 120 + 120 = 150. Step 7: V(15) = 150, meaning after 15 days, the reservoir holds 150 million liters of water. The answer is V(15) = 150; after 15 days, the reservoir contains 150 million liters of water.

4. Hana is tracking the growth of a sunflower in her garden. The height of the sunflower, in centimeters, after t days is modeled by the exponential function H(t) = 40 × 2^(t/4). Hana measures the initial height when she plants the seed at 8:00 AM on day 0. Interpret what H(12) represents in this context, and calculate its value. Answer: H(12) = 320 centimeters; it represents the height of the sunflower 12 days after planting. Solution: The function H(t) = 40 × 2^(t/4) models the sunflower's height in centimeters after t days. H(12) means we substitute t = 12 into the function. Step 2: Calculate the exponent: t/4 = 12/4 = 3.
   Full step-by-step solution

   Step 1: The function H(t) = 40 × 2^(t/4) models the sunflower's height in centimeters after t days. H(12) means we substitute t = 12 into the function. Step 2: Calculate the exponent: t/4 = 12/4 = 3. Step 3: Evaluate 2^3 = 2 × 2 × 2 = 8. Step 4: Multiply by 40: 40 × 8 = 320. Step 5: Interpret the result: H(12) = 320 means that after 12 days from planting, the sunflower's height is 320 centimeters. The answer is H(12) = 320 centimeters; it represents the height of the sunflower 12 days after planting.

5. The function P(t) = 1200(1.08)^t models the population of Kaia's town over t years. What does P(9) represent? Answer: The population after 9 years Solution: The function P(t) = 1200(1.08)^t models population growth over time The initial population is 1200 people The growth rate is 8% per year (from the 1.08 factor) P(9) means we substitute t = 9 into the function This gives us the population after exactly 9 years Therefore, P(9) represents the…
   Full step-by-step solution

   Step 1: The function P(t) = 1200(1.08)^t models population growth over time Step 2: The initial population is 1200 people Step 3: The growth rate is 8% per year (from the 1.08 factor) Step 4: P(9) means we substitute t = 9 into the function Step 5: This gives us the population after exactly 9 years Step 6: Therefore, P(9) represents the population of Kaia's town after 9 years Answer: The population after 9 years

6. Emma is analyzing the growth of a bacteria culture in her biology lab. The population P(t) after t hours is modeled by the function P(t) = 300 × 2^(t/2). If Emma starts with the initial culture at 10:00 AM, at what time will the bacteria population first reach 4,800? Answer: 6:00 PM Solution: Exponential growth models describe how quantities increase rapidly over time. To find when a specific value is reached, you set the exponential expression equal to that value and solve for the time variable.
   Full step-by-step solution

   Exponential growth models describe how quantities increase rapidly over time. To find when a specific value is reached, you set the exponential expression equal to that value and solve for the time variable. The solution will tell you how many hours have passed since the start, which you then add to the initial time to find the actual clock time.

7. Emma is designing a rectangular garden with a fixed perimeter of 60 meters. She wants to maximize the area of the garden. If the length of the garden is represented by x meters, write the area function A(x) in terms of x and determine the dimensions that give the maximum area. Answer: 15 Solution: Let the length be x meters and the width be w meters. The perimeter is 2x + 2w = 60. Solve for w: 2w = 60 - 2x, so w = 30 - x.
   Full step-by-step solution

   Step 1: Let the length be x meters and the width be w meters. Step 2: The perimeter is 2x + 2w = 60. Step 3: Solve for w: 2w = 60 - 2x, so w = 30 - x. Step 4: The area function is A(x) = x * w = x(30 - x) = 30x - x². Step 5: This is a quadratic function A(x) = -x² + 30x, which opens downward. Step 6: The maximum occurs at the vertex x = -b/(2a) = -30/(2*(-1)) = 30/2 = 15. Step 7: The width is w = 30 - 15 = 15. Step 8: The dimensions that give maximum area are 15 meters by 15 meters. The answer is 15.

8. f(x) = 3x² - 5x + 2, find f(4) = ? Answer: 30 Solution: Write the function: f(x) = 3x² - 5x + 2 Substitute x = 4: f(4) = 3(4)² - 5(4) + 2 Calculate the exponent: (4)² = 16 Multiply: 3 × 16 = 48 Multiply: 5 × 4 = 20 Substitute back: f(4) = 48 - 20 + 2 Perform subtraction: 48 - 20 = 28 Perform addition: 28 + 2 = 30 The answer is 30.
   Full step-by-step solution

   Step 1: Write the function: f(x) = 3x² - 5x + 2 Step 2: Substitute x = 4: f(4) = 3(4)² - 5(4) + 2 Step 3: Calculate the exponent: (4)² = 16 Step 4: Multiply: 3 × 16 = 48 Step 5: Multiply: 5 × 4 = 20 Step 6: Substitute back: f(4) = 48 - 20 + 2 Step 7: Perform subtraction: 48 - 20 = 28 Step 8: Perform addition: 28 + 2 = 30 The answer is 30.